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Data analysis is a fundamental aspect of statistics that involves inspecting, cleaning, transforming, and modeling data to discover useful information, suggest conclusions, and support decision-making. In the context of the given exercise, analyzing franchise costs as a percentage of total room revenue for different hotel types requires the use of descriptive statistics, one of which is the boxplot.

A boxplot is a visual representation of the distribution of a dataset. It shows the median, quartiles, and extreme values at a glance, while also indicating the asymmetry, spread, and outliers within the data. The boxplot you create for each hotel type, after ordering the data and calculating quartiles, will provide a quick comparison between the different types of hotels in terms of their franchise cost distribution. This comparative visualization is invaluable, as it can reveal patterns and differences that may not be apparent from a simple list of numbers.

When constructing a boxplot, ensure your data is correct and ordered, as this will allow you to accurately determine key elements such as the median and quartiles. Tailoring the boxplot to highlight interesting data points, such as outliers or specific percentiles, can make your analysis even more comprehensive and insightful for stakeholders.

Calculating quartiles is an essential step in constructing a boxplot and understanding the statistical variability of a dataset. Quartiles divide the data into four equal parts, each comprising 25% of the data points when arranged in ascending order.

To calculate the quartiles:

• First, determine the median (Q2), the middle number of the dataset. If the dataset has an even number of observations, the median is the average of the two middle numbers.
• Then identify the first quartile (Q1), which is the median of the lower half of the data (excluding the median if there's an odd number of observations).
• Finally, find the third quartile (Q3), which is the median of the upper half of the data.

Quartiles are crucial in understanding your data's spread, which is one reason why boxplots are so informative—they visually depict the quartiles, as well as potentially unusual points called outliers, which can strongly influence the interpretation of the data.

Statistical variability, or dispersion, refers to how spread out a set of data is. This is a critical concept when analyzing and interpreting boxplots. A boxplot displays variability through the length of the box, which represents the interquartile range (IQR), and the length of the 'whiskers,' which extend to the smallest and largest data points within 1.5 times the IQR from the quartiles.

Variability can indicate the consistency of data; a narrow box (small IQR) implies that most of the data points are close to each other and the median, while a wide box indicates a greater spread among data points. Additionally, outliers – which appear as individual points beyond the 'whiskers' – can signify extreme variation or data points that deviate significantly from the rest.

When you interpret the boxplots for the different hotel types, consider both the width of the boxes (showing the spread of the middle 50% of data) and the length of the whiskers (reflecting the range of the data). Comparing these elements across the boxplots can provide insights into the relative variability of franchise costs for budget, midrange, and first-class hotels, each of which may have unique operational dynamics and revenue models.